1. Field of the Invention
This invention relates to determining optimal weighting coefficients for a wavelet transform. The invention also is related to a supervised training method to determine the optimal weighting coefficients. The invention further is related to the application of an optimally weighted wavelet transform for detection of microcalcifications in digital mammograms.
The present invention also relates to CAD techniques for automated detection of abnormalities in digital images, for example as disclosed in one or more of U.S. Pat. Nos. 4,839,807; 4,841,555; 4,851,984; 4,875,165; 4,907,156; 4,918,534; 5,072,384; 5,133,020; 5,150,292; 5,224,177; 5,289,374; 5,319,549; 5,343,390; 5,359,513; 5,452,367; 5,463,548; 5,491,627; 5,537,485; 5,598,481; 5,622,171; 5,638,458; 5,657,362; 5,666,434; 5,673,332; 5,668,888; as well as U.S. application Ser. Nos. 08/158,388; 08,173,935; 08/220,917; 08/398,307; 08/428,867; 08/523,210; 08/536,149; 08/536,450; 08/515,798; 08/562,188; 08/562,087; 08/757,611; 08/758,438; 08/900,191; 08/900,361; 08/900,362; 08/900,188; 08/900,192; 08/900,189; and 08/979,639, each of which are incorporated herein by reference in their entirety.
The present invention also relates to technologies referenced and described in the references identified in the appended APPENDIX and cross-referenced throughout the specification be reference to the number, in brackets, of the respective reference listed in the APPENDIX, the entire contents of which, including the related patents and applications listed above and references listed in the APPENDIX, are incorporated herein by reference.
2. Discussion of the Background
Currently, breast cancer is a major cause of death of women in the United States. Clustered microcalcifications are an important indication of early breast cancer because they are present in 30-50% of all cancers found using mammography.
However, despite the above cited methods and improvements, 10-30% of women who have breast cancer and undergo mammography have negative mammograms [12]. In about two-thirds of these missed cases, the radiologist failed to detect a cancer that was evident retrospectively.
To give radiologists a "second opinion" for detection of clustered microcalcifications in mammograms, a computer-aided diagnosis (CAD) scheme based on filtering and feature extraction techniques [3-6] has been proposed. The CAD scheme identifies regions of potential clustered microcalcifications, which are indicated by a marker on the digitized images for review by radiologists.
Many investigators have developed various techniques for the automated detection of microcalcifications. Chan et al. [3,4] reported the efficacy of using a specialized preprocessing step known as the difference-image technique, which is followed by some feature analyses for detection of microcalcifications. Nishikawa et al. [5,6] extended this technique to include additional features such as size and contrast of microcalcifications. Lo et al., [7] Wu et al., [8] and Zhang et al. [9] used artificial neural networks to extract microcalcifications from image data. Other groups have reported applying statistical methods such as Bayesian decision theory [10] to the detection of microcalcifications based on extracted image features such as contrast, shape, and edge ingredients.
Briefly, the wavelet transform is a tool for decomposing images into various scale (size) components. In order to extract and examine the small-scale structures contained in an original image, the wavelet transform uses a fine "probe" that is represented by a small wave. By performing a convolution operation between the small wave and the image, one can extract small structures that have a high correlation with the small wave. The same extraction process can be applied to large-scale structures. In this case, a large (or dilated) wave is used for extraction of large structures.
Let .psi.(x) be a one-dimensional orthogonal wavelet, referred to as a mother wavelet, on the real line IR, and let .phi.(x) be the corresponding scaling function, as constructed by Daubechies [17]. The wavelet .psi..sub.k.sup.j (x) and the scaling function .phi..sub.k.sup.j (x) at scale j and translation k are defined by dilation and translation of the mother wavelet .psi.(x) and the scaling function .phi.(x) as follows: ##EQU1## The separable, two-dimensional wavelet basis ##EQU2## on the real plane IR [2] is defined by tensor products of the one-dimensional wavelets .psi..sub.k.sup.j (x) and the scaling function .phi..sub.k.sup.j (x) as shown below [17]: ##EQU3## Here, the superscripts h, v, and d indicate that the corresponding wavelets are horizontal, vertical, and diagonal wavelets. The subscripts k.sub.1 and k.sub.2 indicate translations along the x and y directions, respectively.
Then the two-dimensional wavelet transform of an input image I(x,y).epsilon.L.sup.2 (IR.sup.2), which is represented as a gray-scale image on a Cartesian coordinate (x,y).epsilon.IR.sup.2, is given by: ##EQU4## Here, L represents the maximum scale of the wavelet decomposition that is equal to log.sub.2 S, with S being the size of the image I(x,y), and mean(I) represents the mean value of the image I(x,y). .PSI..sup.j represents the direct sum of orthogonal projections of I(x,y) onto a detailed space at scale j spanned by ##EQU5## As discussed above, microcalcifications can be detected effectively by more heavily weighting a few low levels (scales around 2) in which microcalcifications are enhanced, and multiplying low weights at the other levels to suppress background structures and noise. For this purpose, we introduce a weight w.sub.j at each scale j in Eq. (1), which yields the following weighted wavelet transform: ##EQU6## The image I.sub.w (x,y) is called a weighted wavelet reconstruction, because it is a reconstructed image from weighted wavelet coefficients.
It should be noted that a more general nonlinear function such as the hyperbola function or the hyperbolic tangent function can be used as a weighting function instead of the weights in the weighted wavelet reconstruction in Eq. (4). However, optimization of these nonlinear weighting functions may be numerically unstable and may result in a suboptimal solution, because they require a number of parameters to specify the shape of the function. On the other hand, optimization of weights is numerically stable and computationally inexpensive, because it contains only a single parameter at each scale to be adjusted.
In a previous study [11], wavelet transforms were combined with the CAD scheme [3,6] to increase sensitivity up to 95% while keeping a low false-positive detection rate of approximately 1.5 clusters per mammogram. In that study, digitized mammograms were first processed by a noise-reduction filter based on mathematical morphology for removal of small spikes [12]. These mammograms were then transformed using the wavelet transform. With its multiresolution capability, the wavelet transform can separate small objects such as microcalcifications from large background structures. It was found that scale 1 (the smallest scale) showed mainly the high-frequency noise included in the mammogram, whereas scales 2 and 3 enhanced microcalcifications effectively. Scales higher than 3 showed a large correlation with the non-uniform background.
Therefore, partially reconstructing mammograms from scales 2 and 3 can effectively enhance microcalcifications, a process called the partial reconstruction method. The microcalcification enhanced images were processed further by the existing procedures in the CAD scheme [3,6], including gray-level thresholding, feature extraction, and clustering tests for detection of clustered microcalcifications. Finally, clusters obtained by the partial reconstruction method were combined with the clusters reported by the difference-image technique through a logical OR operation to yield a final result of clustered microcalcifications.
In the partial reconstruction method, mammograms are reconstructed from the wavelet coefficients in scales 2 and 3, and the coefficients in the other scales are simply eliminated. However, wavelet coefficients in scale 1 may contain some very subtle microcalcifications, and the coefficients in scale 4 or higher may contain some relatively obvious microcalcifications. Therefore, known systems have not formed a substantially optimal combination of weights implementing a weighting function.
Other researchers have investigated the feasibility of utilizing wavelet transforms for enhancement and detection of microcalcifications. Lain et al.[13] performed a scale-dependent enhancement of various mammographic features, including microcalcifications, by selectively scaling and weighting the wavelet coefficients. Qain et al. [14] and Clarke et al. [15] segmented microcalcifications by cascading adaptive spatial filters to perform a tree-structured wavelet transform. Strickland et al. [16] used a bank of multiscale matched filters for detection of microcalcifications. However, modifications of the wavelet coefficients for enhancement of microcalcifications is still an area where improvement is needed.